I know how to compute determinants and I'm familiar with the geometrical meaning of determinant as the scaling factor of a unit (point/square/cube/hypercube)'s area/volume by applying a linear transformation (using a matrix).
However, I have several questions:
- Let's say I define determinant to have the above meaning. How can one derive the formula for computing determinant following just the visual/geometrical meaning?
- Let's say I have an arbitrary closed 2D polytope $P$ and I transform all of its vertices by a matrix $\mathbf{A}$. Is $\det\left(\mathbf{A}\right)$ the scaling factor of polytope's $P$ area after the transformation, i.e. $\det\left(\mathbf A\right) = \frac{P\text{'s area after transform}}{P\text{'s area before transform}}$?
- Imagine I have an open 2D polytope $\overline P$ (which clearly doesn't have any area). How does $\det\left(\mathbf{A}\right)$ relate with the transformed polytope $\mathbf {A}\overline P$?
- Suppose there's a vector $\mathbf x$. What does $\det\left(\mathbf{A}\right)$ say about the transformed vector $\mathbf {Ax}$?